Size and Book to Market Effects: Further Evidence from the French Case

The objective of this paper is to study the market, SMB, HML and the leverage factors in explaining cross-sectional returns. We provide the first empirical analysis of Ferguson and Shockley (2003) theoretical framework on the French stock market. Book to market and size, variables which are correlated with leverage, will appear to explain returns. Our main result is that the leverage factor doesn't subsume the SMB and HML factors. In cross-sectional regressions, only the size premium is statistically significant and help explaining returns. In time-series regressions, the three factors (SMB, HML and leverage), with the market portfolio, do a good job. This result suggests that the leverage portfolio has an additional improvement of the model.


Introduction
The first and the most widely used model of asset pricing is the Capital Asset Pricing Model CAPM ( Sharpe (1964), Lintner (1965), Mossin (1966) and Black (1972)). Because of its simplicity, the beta (β) revolution has had significant impact on the academic and non-academic financial community. The model assumes that investors respect the Markowitz mean-variance criterion in choosing their portfolios.
Its well-known prediction is that the expected excess return on an asset equals the β of the asset times the expected excess return on the market portfolio, where the β is the covariance of the asset's returns with the returns on the market portfolio divided by the variance of the market returns. Roll (1977) argued that the model is not testable because the tests involve a joint hypothesis on the model and the choice of the market portfolio. This problem of joint hypothesis tests was mentioned by other authors ( Ball (1978)).
On the theoretical side, many factor pricing models attempted to explain the cross-section of average asset returns [The Inter-temporal Capital Asset Pricing Model ( Merton (1973)), The Arbitrage Pricing Model ( Ross (1976)) and the intertemporal capital asset pricing model based on consumption ( Rubinstein (1976), Lucas (1978), Breeden (1979) among others)]. However, as Cochrane (2001) has mentioned, all factor models are consumption-based models.
Nevertheless, many patterns emerge from empirical studies which are not explained by the CAPM. Stocks with high earnings to price ratio have, on average, higher returns than stocks with low ratio ( Nicholson (1960) and Nicholson (1968), Basu (1977)). Litzenberger and K. (1979) pointed out a linear positive relation between the dividend yield and the stock returns. Small capitalizations have higher expected returns than big ones ( Banz (1981), Reinganum (1981), Basu (1983)). Schwert (1983) 1 offered a summary of empirical studies on the size effect. There is a positive relation between the level of debt and stock returns ( Bhandari (1988)) and the book to market ratio is considered as an explanatory variable in stock returns ( Stattman (1980), Fama and French (1991), Fama (1991), Chan et al. (1991) and Fama and French (1992)). However, there exist many empirical studies that give contradictory conclusions about these anomalies. Grauer (1999) show that neither the least square method nor the generalized square method can help to know if the mean-variance model is true or false a posteriori 2 .
In our study, we test two hypothesis on the size and the book to market effects on The French Stock Market over July 1976 to June 2001 period.
The first hypothesis is the three factor model of Fama and French (1993). The two authors argue that HM L and SM B portfolios, with the market portfolio, do a good job in explaining cross-sectional returns. Their model summarizes earlier empirical observations and results. However, it is much debated: To be a compensation for risk in a multi-factor version of Merton's (1973) Inter-temporal Capital Asset Pricing Model (ICAPM) or Ross's (1976) Arbitrage Pricing Theory (APT), factors must be related to state variables which justify a risk premium.
The second hypothesis is the proposition of Ferguson and Shockley (2003). The two authors show that the CAPM doesn't work because, in empirical studies, we use an equity-only proxy for the true market portfolio and we ignore the debt claims.
Book to market and size, variables which are correlated with leverage, will appear to explain returns.
Our primary contribution is to provide the first empirical analysis of Ferguson and Shockley (2003)  In the next section, we give a brief summary of the theoretical framework of our study. Methodology used and database considered are discussed in the second part of the paper. In sections three and four, we summarize results and then we conclude.

Theoretical Framework
2.1. The Three Factor Model. Summarizing earlier empirical observations and results, Fama and French (1993) argue that size and book to market ratio are factors of risk that we must remunerate. The unconditional version 3 of the model is expressed in the equation 2.1. Indeed, on the basis of two criteria, size and book to market (BE/ME), Fama and French construct twenty five portfolios, from a sample of the stocks of the NYSE, AMEX and NASD over 366 months (From June 1963to December 1993.
Because monthly stock returns of stocks of small capitalization and high book to market ratio are, on average, higher than these of big capitalizations and low book to market ratio, the two authors propose the following regression 2.2: The results show that the coefficient α i is negative for portfolios located in the extreme quantiles of the stocks of small capitalizations and low ratio book to market and positive for portfolios located in the extreme quantiles of the stocks of big capitalizations and high book to market ratio. In addition to these results on the extremes, the coefficient α i is not significantly different from zero; which makes it possible to affirm that the three factor model explains cross-section stock returns.
Financial literature focusing on explications of size and book to market effects is very large are rich. We limit the presentation here to the main propositions on this subject. ( Lakonishok et al. (1994) and MacKinlay (1995)) argue that the premium of the financial distress is irrational. First of all it can express an over-reaction of the investors. Second, stock returns of firms with distressed financial situation are low, not necessarily during periods of low growth rate of Gross National Product 4 or of low returns of all stocks. Lastly, diversified portfolios of stocks with, as well high as low, ratio book to market; have the same variance of returns.
Other researchers documented other arguments 5 which are inconsistent with the premium of the financial distress: (a) Survivor bias ( Kothari et al. (1995)): But it should be noticed that even if the critic of the survivor biais is true, it is not necessarily in favor of the CAPM ( Kim (1997), Barber and Lyon (1997)). (b) Data-snooping ( Black (1993b), Black (1993a), Lo and MacKinlay (1990)): An extrapolation of data can lead to false conclusions, so how we need the out-ofsample tests. Fama and French (1996b) and Fama and French (1996a) reject this biais 6 . Moreover, the relation between stock returns and the book to market ratio was confirmed by: Davis (1994) on data over a long period; Chan et al. (1991) on Japanese data and Barber and Lyon (1997) on data on the financial institutions 7 , among others. (c) Bad market proxies: Indeed, according to this argument, the model of asset pricing to be retained is that of the CAPM and because we don't know the market portfolio we have anomalies. This is why, the "real" βs are not observed. This problem is called errors-in-variables ( Kim (1997) Daniel and Titman (1997). Indeed, Daniel and Titman give a different interpretation for the relation between book to market ratio and stock returns. They reject the assumption of "factor of risk" in favor of the model of "the characteristics of the firm": A low book to market ratio, which is one of the characteristics of the large firms, causes a low stock returns which does not, necessarily, correspond to a risk. Daniel and Titman (1997) reject the factor model for the U.S. stocks. However, Davis et al. (2000) show that this interpretation is specific to the period of study and confirm the results of the three factor model. In the same way, Lewellen (1999) confirms the superiority of the model of Fama and French (1993) compared to the model of Daniel and Titman (1997) in explaining time-varying expected returns on the U.S. market. Daniel et al. (2000) replicate the Daniel and Titman tests on a Japanese sample and fail to reject the characteristic model 8 . As Fama and French(1993), we make two classifications. We can ask about the significance of a book to market classification? Indeed, a simple understanding of a low book to market ratio is that the market value of the firm is high relative to its book value. This is the case of firms with high growth investment opportunities. Another possible explanation is the existence of intangible assets, like investments in research and development. We mention also the case of firms with low risk with can be expressed in a high market value. Nevertheless, the understanding of the book to market ratio must be made in a context of three dimensions: the life-cycle of the firm, the sector and the stock market.
A size classification: The stocks are grouped in two classes; the stocks of small capitalizations and these of big capitalizations. We consider the capitalization 11 of June of year (t) for the formation of portfolios for the period from July of year (t) to June of year (t + 1). Unlike Fama and French who used the median NYSE size to split NYSE, Amex and NASDAQ stocks (that's why the two size groups contain disproportionate numbers of stocks), we use the median size of the whole sample to make our classification.
In financial literature, many authors ask about the variable to use in making the size classification. In empirical studies, it is usual to consider the market value.
Nevertheless, this variable is subject of debate 12 . As we have mentioned earlier, in our study we consider the market value for the size classification. A firm is classified in small capitalisation for different raisons. We summarize all possible explanations in three categories. First, we have small firms because of their sector of activity.
Second, firms in the beginning of their life-cycle can be classified, temporary, in small capitalizations. Finally, we have destressed firms.
The splits (three book to market groups and two size groups) are arbitrary. However Fama and French (1993) argued that there is no reason that tests are sensitive to this choice. Six portfolios (HS, HB, MS, MB, LS, and LB) are formed with the intersection of the two preceding classifications, made yearly and independently. The monthly returns of each portfolio corresponds to the value-weight monthly returns of the stocks assigned to the portfolio: In our study, the risk free interest rate used is the monthly equivalent rate to: Short term interest rate for the period from July 1976 to January 1981, Money market, one month, rate from February 1981 to January 1987, PIBOR from February 1987 to December 1998 and EURIBOR from January 1999 to June 2001. Table 1 shows that the portfolios in the smallest size quintile and the lowest book to market quintile and these in the biggest size quintile and the highest book to market quintile contain, on average, less stocks than other portfolios. Like table 1 in Fama and French (1993), in the smallest (biggest) size quintile, the number of stocks increases (decreases) from lower to higher book to market portfolios.  This is large compared to Fama and French (1993) in the US-case (only 0.43% with 1.76 standard errors from zero) and Molay (2001) Table 2 summarizes some characteristics of the 16 portfolios, considered as dependent variables in the time series regressions. It shows that the portfolios in the smallest size quintile and the lowest book to market quintile and these in the biggest size quintile and the highest book to market quintile contain, on average, less stocks than other portfolios. Like table 1 in Fama and French (1993), in the smallest (biggest) size quintile, the number of stocks increases (decreases) from lower to higher book to market portfolios.
[Insert table 3 here] The average excess returns of the 16 stock portfolios considered range from 0.81% to 2.71% per month. The positive relation between average excess returns and book to market equity is confirmed. For every size class, average returns of high book to market group are higher than these of low book to market group 17 . Like Molay (1999), in every book to market quintile, average excess returns of small capitalizations are higher than these if big ones. This observation confirms the evidence that there is a negative relation between size and average return. All excess returns of portfolios have high standard deviations (greater than 6% per month). All portfolios produce average excess monthly returns that are more than two standard errors from zero.
On the basis of the adjusted R 2 criterion, we can affirm that the three factor model captures common variation in stock returns 18 . Indeed, for the sixteen portfolios, we obtained an average adjusted R 2 about 68.5%. The market βs are all more than 9 standard errors from zero and adjusted R 2 ranges from 52.0% to 85.7%. Moreover, HM L slopes are related to book to market ratio. For all size classes, they increase from negative values for the lowest book to market quintile to positive values for the highest book to market quintile. Their t-statistics are greater than two, in absolute value, in seven cases. Similarly, SM B slopes are related to size. In every book to market quintile, they decrease from positive values with small capitalizations to negative values with big class. They are more than two standard errors from zero, in absolute value, in thirteen cases out of sixteen. Fama and French (1993) argue that the multi-factor asset pricing models of Merton (1973) and Ross (1976) (2003) propose two mesures to capture the missed beta risk. The first portfolio is based on the ratio debt to equity and it is associated to relative leverage. The second one, based on Altman's Z-score, is used to express the relative distress. As the two authors mentioned, this distinction between relative leverage and relative distress is important. We are aware about that. Nevertheless, the database enables us to construct only the leverage portfolio.   Using least squares and White heteroskedasticity consistent standard errors and covariance, we regress monthly returns of the 16 portfolios according to:  The second hypothesis to test is the marginal contribution of the leverage portfolio in the time-series regressions after controlling for the portion related to SM B and HM L. Table 9 summarizes the results of this test. In the same way, using least squares and White heteroskedasticity consistent standard errors and covariance, we regress monthly returns of the 16 portfolios according to: show that the leverage portfolio give additional improvement to three factor model.
The plotted points are more tighten around the diagonal.

Discussion and Conclusions
Financial literature provides many explanations to size and value premiums. Unlike Fama and French (1993), authors of the three factor model, who argue that market, SM B and HM L portfolios explain stock returns, Ferguson and Shockley (2003) discuss in depth the question of the true market portfolio. They provide a theoretical framework explaining that the size and book to market effects are due to bad market proxies. As long as the market proxy doesn't incorporate the economy's debt claims, these effects are expected to be found in any database. HML and SMB portfolios are related to the missing factor of risk. They loose any explanatory power in the presence of the leverage and distress portfolios.
In this paper, our primary contribution is to provide an empirical analysis of One limit of our study is the use of only one mesure for the firm's leverage. The question is to ask about the additional improvement of the results with a portfolio based on the relative distress. I don't think that such a portfolio will change dramatically the results. Our argument in favor of a such affirmation is the little change in the estimated per unit HM L risk after adding the leverage mesure, despite the high correlation between the leverage and HM L portfolios.

Footnotes
[1] Schwert (1983): "The search for an explication of this anomaly has been unsuccessful. Almost all authors of papers on the 'size effect' agree that it is evidence of misspecification of the capital asset pricing model, rather than evidence of inefficient capital markets. On the other hand, none of the attempts to modify the CAPM to account for taxation, transaction costs, skewness preference, and so forth have been successful at discovering the 'missing factor' for which size is a proxy. Thus, our understanding of the economic or statistical causes of the apparently high average returns to small firms' stocks is incomplete. It seems unlikely that the 'size effect' will be used to measure the opportunity cost of risky capital in the same way the CAPM is used because it is hard to understand why the opportunity cost of capital should be substantially higher for small firms than for large firms." [2]The question of anomalies is much debated in finance. Nevertheless, as Kuhn (1962) said: "Discovery commences with the awareness of anomaly, i.e., with the recognition that nature has somehow violated the paradigm-induced expectations that govern normal science. It then continues with a more or less extended exploration of the area of anomaly. And it closes only when the paradigm theory has been adjusted so that the anomalous has become the expected.".
[3] The conditional version of the model authorizes a temporal variation of the rate of stock returns and coefficients of the factors of risk.
[4] Gross National Product: Chen (1991) indicate that the expected stock returns are negatively correlated with the present rate of growth of GNP and positively correlated with its future rate of growth.
[5] we limit the presentation to three biais related to the use of the data but there exists others; such as errors of corresponding market and accounting data or look ahead bias.
[6] Fama and French (1996b) and Fama and French (1996a) give four arguments: the premium of the financial distress is not special to a particular sample since it is checked for different periods. It was also the subject of many studies made on international database. The size, book to market equity, earning to price and cash flow ratios, indicators of expected incomes (Ball 1978), have a great utility to test models of asset pricing like the CAPM. And in fourth point, the limited number of the anomalies excludes the assumption ofdata-mining.
[7] Barber and Lyon (1997) confirmed the relation between the size, the book to market ratio and the stock returns, published by Fama and French (1992), for the financial institutions (Fama and French considered only the non-financial firms).
[10] Market value to Book divides the Market Value by the Net Book Value (Net Tangible Asset). For companies which have more than one classe of equity capital, both market value and net tangible asset are expressed according to the individual issue.
[11]Market Value is defined as the share price multiplied by the number of ordinary shares issue. The amount in issue is updated whenever new tranches of stock are issued or after a capital change.
[13] R p,t = n i=1 ω i,t * R i,t . Where: R p,t : is the value-weight monthly return of portfolio p in month t.
R i,t : is the monthly return of stock i of portfolio p in month t.
ω i,t : is the ratio of market value of stock i on total value market of portfolio p in month t.
n: is the number of stocks of portfolio p.
[14] Molay (1999) documented an average excess return for the market portfolio of only 0.31%.
[15] Heston et al. (1999) study the case of France (among 12 European countries) for the period from 1978 to 1995. There sample has 418 stocks.
[16] Molay (1999) documented that this negative correlation between SM B and market portfolio can be explained by the fact that market portfolio is value weighted.
When we consider an equal weighted portfolio, this correlation become positive (and it is about 0.13 in Molay's study).
[17] In a first publication on the French market (204 stocks) for the period from July 1992 to June 1997, Molay (1999) confirms the negative relation between size and average return, however he does not found any relation between book to market ratio and average return. Standard deviation of excess stock portfolio returns in his study are less than these of our sample. Molay (2001) considers the period from July 1988 to June 1998 (120 months) for an average of 250 stocks and he confirmed the negative size/average returns relation for only high book to market classes and the positive book to market/average returns relation for only small capitalizations.
[18]For further results on the comparaison between the three factor model and the CAPM, see Lajili (2002) and Lajili (2003b).
[19] Molay (1999) and Molay (2001), obtained two regressions of the three factor model out of nine where intercepts are more than two standard errors from zero.
[20]Stocks with negative borrowing ratio are eliminated. The sample is composed of 636 French stocks. The six size-book to market portfolios are formed from independent sorts on book to market and size as described in the text.
The   The sample is composed of 636 French stocks. The sixteen size-book to market portfolios are formed from independent sorts on size and book to market ratio. The monthly returns of each portfolio corresponds to the value-weight monthly returns of the stocks: We have three explanatory variables: Market, HM L and SM B, as described in   The table presents, for each portfolio, the slopes and their t statistics (between brackets), the adjusted R 2 and the statistic of Durbin-Watson of the market, HM L and SM B

Book to market
portfolios time-series regressions on leverage portfolio (L). All portfolios are described in the text. Using least squares and White heteroskedasticity consistent standard errors and covariance, we regressed monthly excess returns according to:      The following table gives the slopes and their t statistics (between brackets), the adjusted R 2 and the statistic of Durbin-Watson of the 16 time-series regressions. Using least squares and White heteroskedasticity consistent standard errors and covariance, we regress monthly returns of the 16 portfolios according to: